, then you may be the proud owner of counterfeit chips.
I see no load resistor. Try adding one, 100 to 1K Ohms, from output to GND and measure the output voltage again. The regulator could be oscillating.
Make C1 in your diagram 10uF.
a consistent 3V even with a load resistor
Try to replace battery in the multimeter. Or better try measurements with another multimeter. Maybe this one is somehow defective when measuring lower range voltage. I have never experience any of 7805 are fake no matter on sellers price, anyway these regulators are dirty cheap.Do you have any older 7805, 78L05 or even board with? Can you test it with the same multimeter? Indeed, otherwise you may got a stock of fake ICs.